International Journal of Heat and Mass Transfervolkov.eng.ua.edu/Publications/2017-IJHMT-Volkov...48 A.N. Volkov, F. Sharipov/International Journal of Heat and Mass Transfer 114 (2017)

International Journal of Heat and Mass Transfer 114 (2017) 47–61

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International Journal of Heat and Mass Transfer

journal homepage: www.elsevier .com/locate / i jhmt

Flow of a monatomic rarefied gas over a circular cylinder: Calculationsbased on the ab initio potential method

http://dx.doi.org/10.1016/j.ijheatmasstransfer.2017.05.1270017-9310/� 2017 Elsevier Ltd. All rights reserved.

⇑ Corresponding author.E-mail addresses: [email protected] (A.N. Volkov), [email protected]

(F. Sharipov).

Alexey N. Volkov a,⇑, Felix Sharipov b

aDepartment of Mechanical Engineering, University of Alabama, Hardaway Hall, 7th Avenue, Tuscaloosa, AL 35487, USAbDepartamento de Física, Universidade Federal do Paraná, Caixa Postal 19044, Curitiba 81531-990, Brazil

a r t i c l e i n f o a b s t r a c t

Article history:Received 21 August 2016Received in revised form 18 March 2017Accepted 29 May 2017

Keywords:Ab initio potential methodDSMC methodCylinder in cross-flowTransitional flow regime

Two-dimensional flows of argon and helium over a circular cylinder are calculated by the DirectSimulation Monte Carlo (DSMC) method in the range of the free stream Mach number Ma1 from 0.5to 10 in nearly free molecular, transitional, and nearly continuum flows. In DSMC simulations, inter-particle collisions are calculated with the ab initio (AI) potential method based on the interatomic inter-action potentials established in quantum mechanical calculations. It is shown that the AI potentialmethod enables computationally efficient simulations of multidimensional rarefied gas flows withoutintroducing semi-empirical models of collision cross sections. The calculated values of the drag CD andheat flux CQ coefficients of the cylinder for Ar and He at the same values of Ma1, rarefaction parameter,and surface-to-free-stream temperature ratio are found to be different in less than 1%, ensuring smallsensitivity of CD and CQ to the species of a monatomic gas. The simulation results obtained with the AIpotential method are systematically compared with results obtained with the hard sphere (HS) molecularmodel. It is found that the choice of the HS diameter based on the condition of the identical viscosity ofthe real and HS gases at the free stream temperature ensures calculations of CD and CQ in sub- and super-sonic flows at Ma1 6 2 with errors less than 3% and 6.5%, correspondingly. For hypersonic flows, thischoice of the HS diameter is unsatisfactory and results in the errors up to 7% in CD and 28% in CQ . Asemi-empirical rule that defines an optimum HS diameter in super- and hypersonic flows is suggested.With the use of this rule, the HS model is capable of predicting CD and CQ with errors less than 1% and3%, correspondingly, and also provides a good accuracy in calculations of local flow parameters.

� 2017 Elsevier Ltd. All rights reserved.

1. Introduction

The Direct Simulation Monte Carlo (DSMC) method [1] cur-rently is the major numerical tool for computer modelling of rar-efied gas flows. It is a particle-based simulation method, where agas flow is represented by a set of simulated particles, which par-ticipate in binary collisions with each other and move freelybetween collisions. The core of the DSMC method is a procedurefor stochastic sampling of binary collisions between simulated par-ticles, which is derived in agreement with the collision term in theBoltzmann kinetic equation [2]. In order to apply this method forpractical problems, one needs to use a molecular model thatdefines the collision frequency and post-collisional velocity of sim-ulated molecules. Such a molecular model is usually formulated interms of the collision cross sections [1]. For monatomic gases, the

collision cross section can be deduced from the known potentialcurve, describing the interaction energy of a pair of molecules asa function of distance between them, e.g., [1,3]. At the same time,the collision cross sections traditionally used in DSMC simulationsare formulated in the form of semi-empirical models, which aredeveloped to fit experimental values of gas viscosity and othertransport properties at a particular temperature. The hard sphere(HS) and variable hard sphere (VHS) models are among the mostpopular molecular models for DSMC simulations of non-reactivegas flows [1]. Since all such molecular models are approximative,the degree of their accuracy must be verified against models thataccurately describe mechanics of inter-particle collisions.

Recently, a method enabling the use of arbitrary interatomicpotentials for DSMC simulations of monatomic gas flows wasdeveloped in Ref. [4] and then applied to study a number of rar-efied gas flows [5–7]. In this approach, the post-collisional veloci-ties of simulated molecules are defined based on the preliminarytabulated dependence of the deflection angle [1] on the energy ofcolliding particles and geometric impact parameter [8]. This

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48 A.N. Volkov, F. Sharipov / International Journal of Heat and Mass Transfer 114 (2017) 47–61

approach can be used for arbitrary potentials, including the semi-empirical Lennard-Jones potential and ab initio potentials estab-lished in quantummechanical calculations, e.g., [9,10]. Hereinafter,this approach is termed the ‘‘ab initio (AI) potential method,” sinceit is capable of calculating the post-collisional velocities from thefirst principles. A similar approach for calculations of binary colli-sions based on the Lennard-Jones potential was suggested in Ref.[11], where the dependence of the deflection angle on the energyand impact parameter is approximated by high-order polynomials.It is worth noting, however, that the representation of the deflec-tion angle in the AI potential method [4] in the tabulated formenables faster sampling of post-collisional velocities and requiresonly a tiny fraction of computer memory used in typical DSMCsimulations.

The goal of the present paper is threefold. First, we demonstratethat the AI potential method developed in Ref. [4] can be efficientlyused for DSMC simulations of multidimensional rarefied gas flowsover bodies in a broad range of flow conditions, from sub- tohypersonic velocities and from the nearly free molecular to nearlycontinuum flow regimes. For this purpose we performed system-atic simulations of rarefied gas flows over a circular cylinder basedon the AI potential method for argon and helium. Second, we com-pare the flows of argon and helium calculated based on the AIpotential method in order establish how the local flow parametersin reduced units as well as the drag and heat flux coefficients of thecylinder in cross-flow depend on the gas species and free streamtemperature. Finally, we compare the results obtained based onthe AI potential method with results based on the simple androbust HS model in order to quantify the errors introduced bythe application of the HS model in calculations of the drag and heatflux coefficients. We chose only the HS model for comparison withthe AI potential method, because the HS model is known to be thesimplest molecular model for kinetic simulations, it was used inmultiple theoretical and numerical studies, and its predictions, in

Fig. 1. Sketch of Cartesian coordinates Oxy and computational domain ABCDEF (a), asmacroscopic gas parameters (c) in simulations of two-dimensional flow over a circular

reduced units, do not depend on the gas species. The last propertymakes the HS model particularly attractive for multiple problemsin aerothermodynamics.

The flow over a cylinder in the transitional flow regime is one ofthe major problems that are traditionally used for testing newnumerical approaches and molecular models in the rarefied gasdynamics. This flow was a subject of multiple theoretical, e.g.,[12–21], and experimental, e.g, [22,23], studies. The kineticsimulations of flow over a cylinder were performed using theDSMC method [12,14,16,18,20] and direct numerical solutionof the Bhatnagar-Gross-Krook model kinetic equation[13,15,17,19,21].

Our main finding is that the local distributions of gas parame-ters in reduced units as well as the drag and heat flux coefficientsof the cylinder are marginally affected by the monatomic gas spe-cies and dimensional free stream temperature in the whole rangeof flow conditions under consideration. We also found a simplerule that defines the HS molecular diameter and ensures smallerrors in calculations of the drag and heat flux coefficients basedon the HS model.

The paper is organized as follows. In Section 2, a kinetic modelof monatomic gas flow over a cylinder is described. The criteria offlow similarity are formulated in Section 3. The algebraic relation-ships for the cylinder drag and heat flux coefficients in the freemolecular flow regime are presented in Section 4. The DSMCmethod, its numerical parameters, and validation of the computa-tional code are discussed in Section 5. Then, in Sections 6–8, theresults obtained for sub-, super-, and hypersonic flows based onthe AI potential method and HS molecular model are describedand compared with each other. The conclusions for all flowregimes under consideration are presented in Section 9. Finally,Appendix A contains data on comparison of the CPU times requiredfor the DSMC collision sampling based on the HS model and AIpotential method.

well as computational meshes used for collision sampling (b) and calculation ofcylinder. The plane y ¼ 0 is assumed to be the flow plane of symmetry.

A.N. Volkov, F. Sharipov / International Journal of Heat and Mass Transfer 114 (2017) 47–61 49

2. The mathematical model

A planar two-dimensional flow of a monatomic gas over a circu-lar cylinder of radius R is considered in an inertial frame of refer-ence, which is at rest with respect to the cylinder center O(Fig. 1). A Cartesian coordinate system x; y with the basic unit vec-tors ex and ey is introduced, where the axis Ox directs along the gasvelocity vector in the free stream, U1 ¼ U1ex, and the cylinder axisis perpendicular to the plane Oxy. The position of an arbitrary pointP on the cylinder surface in the plane Oxy is characterized by theangle h and the position vector Rn, where n ¼ cos hex þ sin hey isa unit vector that is normal to the cylinder surface and directedoutward from the cylinder. The plane y ¼ 0 is assumed to be theflow plane of symmetry, so that the flow is considered only inthe half-space y P 0.

The gas flow is described by a kinetic model based on the Boltz-mann equation [24]

@f@t

þ v � @f@r

¼ IBðf ; f Þ; ð1Þ

formulated in terms of the velocity distribution function f ðr;v ; tÞ ofgas molecules, where r and v are the position and velocity vectorsof a molecule in the considered frame of reference, and t is the time.The function f ðr;v; tÞ is assumed to be normalized by the numberdensity of gas molecules n, i.e. nðr; tÞ ¼ R f ðr;v; tÞdv . The Boltzmanncollision term IBðf ; f Þ in Eq. (1) is represented in the form

IBðf ; f Þ ¼ZZ 2p

0

Z bmax

0f 0f 01 � ff1� �

crbdbd�dv1; ð2Þ

where f ¼ f ðr;v ; tÞ, f 1 ¼ f ðr;v1; tÞ, f 0 ¼ f ðr;v 0; tÞ, f 01 ¼ f r;v 01; t

� �,

cr ¼ jv1 � vj, b is the impact parameter, � is the azimuthal angle,v 0 and v 0

1 are velocities of molecules after a collision with velocitiesv and v1 before the collision at given b and �,

v 0 ¼ 12ðv þ v1 � crd

0Þ; v 01 ¼ 1

2ðv þ v1 þ crd

0Þ; ð3Þ

d0 ¼ d cosvþ t sinv cos �þ d� t sinv sin � ð4Þis the unit vector defining the direction of the relative velocity ofmolecules after the collision, d ¼ ðv1 � vÞ=cr is the unit vectordefining the direction of the relative velocity of molecules beforethe collision, t is an arbitrary unit vector orthogonal tod;v ¼ vðEr; bÞ is the deflection angle that depends on the collisionkinetic energy Er ¼ mc2r =4 and impact parameter b, and bmax is thecutoff value of b for adopted molecular model.

In the present paper, all results are obtained with twoapproaches for calculations of interatomic collisions. The firstapproach is based on the classical model of HS molecules of diam-eter dHS, when v ¼ arcsinðb=dHSÞ; bmax ¼ dHS, and the total collision

cross section is equal to rHS ¼ pd2HS [1]. In the second approach,

the dependence v ¼ vðEr ; bÞ for a particular species of a monatomicgas is determined based on the AI potential method from the solu-tion of the classical trajectory problem for two colliding particleswith given interaction potential. The algorithm of calculations ofvðEr ; bÞ is described in Ref. [3]. According to Ref. [4], the depen-dence of vðEr ; bÞ is used in the form of a two-dimensional table cal-culated for b < bmax and Er < Er;max based on the ab initiointeratomic potentials that were established in quantum–mechan-ical calculations for helium [9] and argon [10]. In this case, the total

collision cross section is equal to rAI ¼ pb2max. In Refs. [9,10], values

of interaction energy in He-He and Ar-Ar pairs as functions of dis-tance between atoms were calculated based on coupled-cluster-type methods and first presented in the tabulated form. Next, thecomputed data were used to fit the modified Tang-Toennies poten-

tial function [25]. The analytical representations of the potentialcurves in the form of the modified Tang-Toennies functions wereused in the present paper in order to calculate vðEr ; bÞ. The limitsbmax and Er;max were determined in preliminary calculations inorder to provide results that are non-sensitive to further increasein bmax and Er;max within a given tolerance. The further details onthe choice of bmax and Er;max are given in Section 5. The CPU timesrequired for the DSMC collision sampling based on the HS modeland AI potential method are compared with each other inAppendix A. Hereinafter, the three types of gases considered inthe present paper are termed as ‘‘AI Ar,” ‘‘AI He,” and ‘‘HS gas”.

Reflection of gas molecules from the cylinder surface isdescribed by the Maxwell model of diffuse scattering [3,24,8]

at v � n > 0 : f ðRn;v ; tÞ

¼ 2p

exp � v2

C2w

!Zv 0 �n<0

jv 0 � njC4w

f ðRn;v 0; tÞdv 0; ð5Þ

where Cw ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffi2RTw

p;R ¼ kB=m is the gas constant, kB is the Boltz-

mann constant, m is the mass of a gas molecule, and the relaxationtemperature is assumed to be equal to the temperature of the cylin-der surface Tw.

In the free stream, the velocity distribution of gas molecules isassumed to be a Maxwellian with given number density n1, veloc-ity U1, and temperature T1:

f ðr;v ; tÞ ! f1ðvÞ at jrj ! 1; ð6Þ

f1ðvÞ ¼ n1

ð ffiffiffiffip

pC1Þ3

exp �ðv � U1Þ2C21

!: ð7Þ

where C1 ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffi2RT1

p.

The goal of calculations in the present paper is to find onlysteady-state flows over the cylinder, which can be obtained in alimit t ! 1 by considering a time-dependent process that evolvesstarting from an arbitrary initial distribution of gas molecules, e.g,uniform distribution f1ðvÞ in the entire domain. Hereinafter,therefore, only the final steady-state distribution function of gasmolecules f ðr;vÞ is used in order to define the aerodynamic dragforce FD of the cylinder and the heat flux Q on the cylinder surfaceper unit length. Values of FD and Q can be found by integrating thedistributions of x-component of the stress vector px and heat fluxdensity q over the cylinder surface [3]

FD ¼ 2RZ p

0pxðhÞdh; Q ¼ 2R

Z p

0qðhÞdh; ð8Þ

where

pxðhÞ ¼ �Z

mðv � exÞðv � nÞf ðRn;vÞdv; ð9Þ

qðhÞ ¼ �Z

mv2

2ðv � nÞf ðRn;vÞdv : ð10Þ

3. Criteria of flow similarity

The goal of this section is to determine the set of criteria of sim-ilarity specific for rarefied gas flows over a circular cylinder consid-ered based on the AI potential method and HS molecular model.The following discussion is focused on the representation of thecylinder drag force and heat flux in reduced units, however, itcan be straightforwardly generalized for any local flow parameteror for other classes of flows.

In aerothermodynamics, the drag force FD and heat flux Q aretraditionally represented in the form of the drag, CD, and heat flux,


CQ , coefficients, e.g., [3,26]. For a cylinder in cross-flow, these coef-ficients are usually introduced as follows

FD ¼ 12CDq1U2

1ð2RÞ; ð11aÞQ ¼ CQq1cpU1T1ð2pRÞ: ð11bÞ

The heat flux coefficient of a blunt body in high-speed flows, inturn, is often represented in the form containing the Stanton num-ber St,

CQ ¼ StTad

T1� Tw

T1

� �; ð12Þ

where Tad is the adiabatic temperature, i.e. such a uniform bodytemperature when Q ¼ 0. Numerical calculation of the adiabatictemperature for given free stream conditions requires multiple sim-ulations with varying Tw, therefore, in the present paper the onlycoefficient CQ was calculated. The dimensionless coefficients CD

and CQ can be represented in the form of functions of dimensionlessparameters that serve as criteria of flow similarity. The set of thesimilarity criteria can be established by re-writing the mathemati-cal model given by Eqs. (1)–(7) in a dimensionless form [27].

3.1. Flow similarity for the AI potential method

In the case of the AI potential method, in order to introducereduced units in the problem given by Eqs. (1)–(7) one needs todefine an appropriate scale for the impact parameter b in the Boltz-mann collision term given by Eq. (2). In the present paper, such alinear scale is introduced as an equivalent diameter d� of HS mole-cules, which provides agreement between viscosities of the AI,lAIðTÞ, and HS, lHSðTÞ, gases at the free stream temperature T1,i.e. based on the condition lAIðT1Þ ¼ lHSðT1Þ. For the AI Ar andAI He, viscosity lAIðTÞ is obtained based on the AI interatomicpotentials using the 5-th order expansion with respect to Soninpolynomials and given in the tabulated form in Refs. [28,29]. Inthe HS gas, viscosity in the infinite approximation of theChapman-Enskog theory is given by the equation [30]:

lHSðTÞ ¼ Cl

ffiffiffiffiffiffiffiffiffiffi2RT

p

d2 ; ð13Þ

where coefficient Cl is numerically found to be equal to 0.126668[31,32], which is only slightly different from the value of5=ð16

ffiffiffiffiffiffiffi2p

pÞ found theoretically in the first approximation of the

Chapman-Enskog theory using the 1-st order expansion withrespect to Sonin polynomials [30]. Thus, the equivalent HS diameterd�, used as a scale for b in the AI potential method, is defined asfollows

d� ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiCl

C1lAIðT1Þ

s: ð14Þ

Then one can introduce the quantities in reduced units as fol-

lows: �r ¼ r=R; �v ¼ v=C1;�f ¼ fC31=n1; �b ¼ b=d�. Solution of the

problem given by Eqs. (1)–(7) in such reduced units depends onthree dimensionless criteria of similarity, including the velocity

coefficient S1 ¼ U1=C1, density parameter D1 ¼ Rd2�n1, and tem-

perature ratio Tw=T1. Instead of S1 and D1, we will characterizethe problem with other parameters, namely, the free stream Machnumber Ma1 and rarefaction parameter d1:

Ma1 ¼ U1ffiffiffiffiffiffiffiffiffiffiffiffifficRT1

p ; ð15aÞ

d1 ¼ Rp1lAIðT1ÞC1

; ð15bÞ

where p1 ¼ n1kBT1 is the pressure in the free stream andc ¼ cp=cv ¼ 5=3 is the ratio of isochoric, cv ¼ 3R=2, and isobaric,cp ¼ 5R=2, specific heats of a monatomic gas. The rarefactionparameter d1 characterizes the degree of rarefaction and for theAI potential method replaces the inverse Knudsen number, the ratioof the flow length scale to the mean free path of gas molecules.Introduction of the rarefaction parameter in the form given by Eq.(15b) allows one to avoid the ambiguity in the definition of themean free path for the AI potential method, which is characterizedby diverging total collision cross section.

The dimensionless solution for the AI potential method dependsnot only on Ma1; d1, and Tw=T1, but also is determined by the gasspecies and dimensional flow temperature T1, because the depen-dence vðEr ; bÞ in the AI potential method can be calculated only fora particular gas species. In the reduced units, it can be written asvðkBT1�c2r =2; d�ðT1Þ�bÞ, where �cr ¼ cr=C1. Thus, any solution of theproblem for a particular AI gas can be obtained only ifMa1; d1; Tw=T1; and T1 are fixed. Correspondingly, the drag andheat flux coefficients also depend on the dimensional temperatureT1:

CD ¼ CD;GasðMa1; d1; Tw=T1; T1Þ; ð16aÞCQ ¼ CQ ;GasðMa1; d1; Tw=T1; T1Þ; ð16bÞwhere the subscript ‘‘Gas” indicates that the drag and heat flux coef-ficients are defined for a particular gas species. The dependence ofCD and CQ on the gas species and T1 is considered in Section 7.

3.2. Flow similarity for the HS model

If the HS diameter dHS is considered as a fixed parameter, thenthe dimensional analysis allows one to conclude that the dragand heat flux coefficients in HS gas flows can be represented inthe form

CD ¼ CDðMa1; d1ðHSÞ; Tw=T1Þ; ð17aÞCQ ¼ CQ ðMa1; d1ðHSÞ; Tw=T1Þ; ð17bÞwhere the rarefaction parameter d1ðHSÞ is defined as

d1ðHSÞ ¼ Rp1lHSðT1ÞC1

; ð18Þ

and lHSðT1Þ is given by Eq. (13) at d ¼ dHS.Eqs. (17) show that CD and CQ in the HS gas are completely

defined by three criteria of similarity. This simple and well-known conclusion, however, does not account for the fact that inthis consideration the HS diameter dHS is treated as a knownparameter. At the same time, in order to make the HS molecularmodel useful, one needs to choose such value of dHS that providesa reasonable agreement between the simulation results and thereal gas flow. The problem of the choice of such dHS is a non-trivial and ill-conditioned problem, since the ‘‘optimum” value ofdHS depends not only on the gas species, but also on the problemunder consideration and even particular flow parameters. In thiscase, it is interesting to pose the question whether it is possibleto find a simple rule defining the optimum dHS, which providesan agreement between the HS and AI gases sufficient for practicalapplication or not? In the present paper, we address this questionfor the cylinder in cross-flow.

In order to connect dHS to particular flow conditions we assumethat dHS is defined by the condition lHSðTref Þ ¼ lAIðTref Þ at some ref-erence temperature Tref . Then dHS is calculated similarly to d� as

dHS ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiCl

ð2RTref Þ1=2lAIðTref Þ

sð19Þ

and Eq. (18) reduces to

Fig. 2. Ratio d1ðHSÞ=d1 given by Eq. (20) versus Tref for argon. Values of lAIðTÞadopted from Ref. [29] are used for calculations. Value of T1 for every curve isindicated in the figure panel.


d1ðHSÞ ¼ d1lHSðTref ÞlAIðTref Þ ¼ d1

lAIðT1ÞlAIðTref Þ

ffiffiffiffiffiffiffiffiTref

T1

s: ð20Þ

The problem of finding the optimum value of dHS, thus, reducesto the problem of finding the optimum Tref . The advantage of thisreformulation is that Tref can be chosen as a certain characteristictemperature that is mainly defined by Ma1; T1, and Tw. This free-dom in the choice of Tref and the use of viscosity for a particular gasin Eq. (20) mean that Eqs. (17) must be replaced by

CD ¼ CD;HSðMa1; d1ðHSÞðTref ;AIÞ; Tw=T1Þ; ð21aÞCQ ¼ CQ ;HSðMa1; d1ðHSÞðTref ;AIÞ; Tw=T1Þ: ð21bÞwhere the parameter ‘‘AI” symbolically indicates that the value ofd1ðHSÞ depends on the gas species which has to be fitted by the HSmodel.

The reference temperature Tref , of course, can be formally cho-sen to be equal to the free stream temperature T1, which ensuresthat d1ðHSÞ ¼ d1, while d1ðHSÞ=d1 – 1 at Tref – T1, as it is illustratedfor the AI Ar in Fig. 2. The choice Tref ¼ T1, however, is not justified,e.g., for hypersonic flows, Ma1 P 4, when the free stream temper-ature does not characterize the thermal regime of the cylinder sur-face. In Sections 7 and 8, a simple rule is suggested in the formTref ¼ Tref ðMa1; T1; TwÞ, which provides good agreement betweensimulations results obtained for the AI and HS gases in a broadrange of flow conditions.

Traditionally, the degree of rarefaction in flows of moleculeswith finite total cross section is characterized by the Knudsennumber, e.g., based on the equilibriummean free path of molecules

in the HS gas, Kn1ðHSÞ ¼ 1=ðffiffiffi2

ppd2n12RÞ. If dHS is defined by Eq.

(19), then the Knudsen number Kn1ðHSÞ is inversely proportionalto d1ðHSÞ:

Kn1ðHSÞ ¼ 1

25=2pCld1ðHSÞ� 0:5628

d1ðHSÞ: ð22Þ

This equation allows one to define the Knudsen number for allsimulation conditions considered in Sections 6–8.

4. Drag and heat flux coefficients in free molecular flow

In steady-state, free molecular flow, the drag and heat flux coef-ficients of any convex body can be easily obtained. In this case, thevelocity distribution function of molecules incident to the body

surface is equal to the Maxwell–Boltzmann distribution in the freestream given by Eqs. (6) and (7). By inserting this function into theboundary condition on the body surface given by Eq. (5), one canfind then the velocity distribution function of molecules reflectedfrom the body in the form of a Maxwellian with zero gas velocity,body temperature Tw, and number density which is given by theimpermeability condition. Then, for a circular cylinder in cross-flow, one can use the distribution functions of incident andreflected molecules in order to calculate the stress vector and heatflux density distributions according to Eqs. (9) and (10), and finally,the drag force and heat flux given by Eqs. (11). This procedureresults in [3,16,33]:

CD ¼ffiffiffiffip

pS1

exp � S212

!S21 þ 3

2

� �I0

S212

!S21 þ 1

2

� �I1

S212

!" #

þ p3=2

4S1

ffiffiffiffiffiffiffiTw

T1

s; ð23Þ

St ¼ 25ffiffiffiffip

pS1

exp � S212

!S21 þ 1� �

I0S212

!þ S21I1

S212

!" #; ð24Þ

Tad

T1¼ S21

2þ 54� 14

I0S212

� �S21 þ 1� �

I0S212

� �þ S21I1

S212

� � ; ð25Þ

where the heat flux coefficient CQ is calculated based on St andTad=T1 according to Eq. (12) and I0ðxÞ and I1ðxÞ are the modifiedBessel functions of the first kind

I0ðxÞ ¼ 1p

Z p

0ex cos sds; I1ðxÞ ¼ 1

p

Z p

0ex cos s cos sds: ð26Þ

Values of CD and CQ calculated for free molecular flow based onEqs. (23)–(25) are given below in Tables 4–6 along with thenumerical results for transitional flows.

5. Numerical method for transitional flows and its validation

Simulations of transitional flows past a circular cylinder wereperformed with a versatile parallel DSMC code previously devel-oped for simulations of laser-induced plume expansion [34] andlater applied for a number of problems including vapor jet incross-flow [35] and flow over a spinning sphere [36]. In this code,the intermolecular collisions are sampled at a uniform Cartesiangrid based on the ‘‘No Time Counter” (NTC) method [1] and thecut-cell approach, e.g., [37], which is used to accommodate the gridto the flow boundaries of arbitrary geometry. Sampling of gasmacroscopic parameters is performed on additional surface andvolume structured grids that are uniform in cylindrical coordinateson the plane Oxy. These grids are schematically shown in Fig. 1. Forthe present study, the computational code was redesigned in orderto enable simulations based on the AI potential method. For thevalidation of the redesigned code, it was first applied to the prob-lem of the heat conduction in a monatomic rarefied gas betweentwo parallel planes under conditions studied in Ref. [7]. It wasfound that the difference between the heat fluxes obtained withthe present code and in Ref. [7] is within 0.5%. The code was alsoapplied for simulations of free molecular flows past a circularcylinder, and agreement within 0.1% with CD and CQ calculatedbased on Eqs. (12) and (23)–(25) was established in the range ofMa1 from 0:5 to 10.

A systematic study of the dependence of CD and CQ on multiplenumerical parameters of the problem were performed next. Thenumerical parameters include the sizes of the computationaldomain, L1; L2, and H (Fig. 1(a)), uniform cell size Dx, number of

Fig. 3. Fields of number density n=n1 of the HS gas at Ma1 ¼ 0:5; d1 ¼ 1, andTw=T1 ¼ 1: (a) L1=R ¼ L2=R ¼ 10;H=R ¼ 5; (b), L1=R ¼ L2=R ¼ 40;H=R ¼ 10.


simulated particles per cell in the uniform free stream Nc , time stepsize Dt, the time interval required to reach the steady state fromthe initial uniform distribution of gas molecules in the entiredomain tss, the number of time steps Ns, used for sampling of gasparameters, and the cutoff values Er;max and bmax for tabulateddependencies v ¼ vðEr ; bÞ. The optimal choice of these numericalparameters, ensuring the calculations of CD and CQ with a desiredaccuracy without using excessive computational resources, is dif-ferent depending on the flow Mach number Ma1. The dependen-cies of CD and CQ on numerical parameters at Ma1 ¼ 2 areillustrated with the data given in Table 1. All simulations in Table 1are performed with the time step size Dt corresponding toDtv�=Dx ¼ 0:05, where v� ¼ U1 þ C1. It was found that the reduc-ing the time step in five times results in the changes of CD and CQ inless than 0.5% with respect to values listed in the table. Suchdependencies are typical for the considered range of Ma1 from0:5 to 10 with an exclusion related to the domain size in subsonicflows. In subsonic flows, the disturbances introduced by thestreamlined body propagate far from the body in all directions,so that accurate simulations of subsonic flows with boundary con-ditions far from the cylinder given by Eq. (5), require large compu-tational domains with L1; L2;H � 40, while for super- andhypersonic flows such large domains are redundant. The exces-sively small size of the computational domain in simulation of sub-sonic flows may result in moderate errors in CD and CQ , but leads tothe strong distortion of the flow field, as it is illustrated in Fig. 3.

As a result of this study, we chose such values of numericalparameters, which ensure that the numerical errors in CD and CQ

are less than 1%. The summary on the choice of numerical param-eters for three values of Ma1 equal to 0:5;2, and 10 consideredbelow in Sections 6–8 is given in Table 2. The chosen values ofthe energy cutoff parameter Er;max (not listed in Table 2) for bothargon and helium provide negligible number of ”invalid” collisionswith Er > Er;max at Ma1 6 2 (typically, only a few such collisionswere registered in an individual simulation). In hypersonic flowat Ma1 ¼ 10, the chosen value of Er;max ensures that less than0.1% of the total number of collisions occurs at Er > Er;max.

In order to validate the computational code and our choice ofnumerical parameters, flows past a cylinder were calculated underconditions identical to conditions considered in Refs. [16,18]. Thesimulation in Ref. [16] were performed for helium based on theVHS molecular model [1] with the viscosity index 0:66 and refer-ence diameter 2:33 � 10�10 m at temperature of 273 K. AtMa1 ¼ 2:9, Knudsen number Kn1 ¼ lmfp;1=ð2RÞ ¼ 0:2 andTw=T0 ¼ 1:66, where lmfp;1 is the equilibrium mean free path ofthe VHS molecules at temperature T1 and T0 is the adiabatic stag-nation temperature given below by Eq. (29), the DSMC simulationsin Ref. [16] resulted in CD ¼ 1:92 and CQ ¼ �0:318. It is worth not-ing that, in Ref. [16], the definition of the heat flux coefficient is dif-

Table 1Drag, CD , and heat flux, CQ , coefficients for the HS gas at Ma1 ¼ 2; Tw=T1 ¼ 1, and d1 ¼ 10computational mesh Dx, number of simulated molecules in a cell located in the free stredomain boundaries (Fig. 1(a)). Numerical parameters corresponding to case IX are adopte

Case Dx=R Nc L1=R

I 0.2 10 5II 0.1 10 5III 0.05 10 5IV 0.02 10 5V 0.01 10 5VI 0.02 30 5VII 0.02 10 5VIII 0.02 10 5IX 0.02 30 5X 0.02 90 5XI 0.02 10 5

ferent from the definition adopted in the present paper, so that thevalue of CQ ¼ �0:318 calculated in accordance with Eqs. (11b) and(12) was provided by M. Yu. Plotnikov. For the purpose of compar-ison, we performed DSMC simulations with the same molecularmodel and flow parameters using the numerical parameters indi-cated in Table 2 for Ma1 ¼ 2 and d1 ¼ 10 and obtainedCD ¼ 1:91 and CQ ¼ �0:315. These values are only 0.5% and 0.9%different from values reported in Ref. [16].

Simulations in Ref. [18] were performed for argon based on theVHS molecular model [1] with the viscosity index 0:734 and refer-ence diameter 3:595 � 10�10 m at temperature of 1000 K, forMa1 ¼ 10; T1 ¼ 200 K, and a 12-in.-diameter cylinder with sur-face temperature Tw ¼ 500 K. For comparison, we chose the caseof Kn1ðHSÞ ¼ 0:05, when the total drag is equal to FD ¼ 8:91 N m�1

or, according to Eq. (11a), CD ¼ 1:495. For the purpose of compar-ison, we performed DSMC simulations with the same molecularmodel and flow parameters using the numerical parameters indi-cated in Table 2 for Ma1 ¼ 10 and d1 ¼ 10 and obtainedCD ¼ 1:479, which is 1.1% different from the value calculated basedon results reported in Ref. [18] (Table 3). We also found that thepeak heat transfer rate q0=q1 in the stagnation point at the cylindersurface in our simulations is different from the value reported inRef. [18] in 5.1%. Based on results of this comparison we concludedthat the adopted values of numerical parameters in Table 2 provideagreement in the integral parameters of the cylinder drag and heattransfer within 1% compared to data reported in Refs. [16,18]. Inthe hypersonic flow, the agreement between local heat fluxes in

at various numerical parameters of the DSMC method, including the cell size of theam Nc , and geometrical parameters L1 ; L2, and H, characterizing the positions of thed for further simulations at Ma1 ¼ 2.

L2=R H=R CD CQ

10 5 1.776 0.129710 5 1.688 0.108210 5 1.665 0.098210 5 1.651 0.093610 5 1.647 0.092710 5 1.649 0.093315 5 1.650 0.093510 10 1.641 0.094110 10 1.641 0.093810 10 1.640 0.093810 15 1.641 0.0940

Table 2Numerical parameters, including the cell size of the computational mesh Dx, number of simulated molecules in a cell located in the free stream Nc , time step size Dt, andgeometrical parameters L1; L2, and H, characterizing position of the domain boundaries (Fig. 1) adopted for the DSMC simulations at various Ma1 and d1 . Here v� ¼ U1 þ C1 .

Ma1 d1 Dx=R Nc Dtv�=Dx L1=R L2=R H=R

0.5 0.1–10 0.02 10 0.025 35 35 102 0.1–10 0.02 30 0.05 5 10 102 30 0.01 30 0.1 5 10 1010 0.1–10 0.02 10 0.05 5 10 1010 30 0.01 10 0.1 5 10 10

Table 3Drag coefficient CD , heat flux coefficient CQ , and the heat flux in the stagnation point q0=q1 obtained for Ar in Ref. [18] and in the present paper based on the VHS molecular modelwith parameters x ¼ 0:734; d ¼ 3:595 � 10�10 m at T ¼ 1000 K adopted from Ref. [18], for the AI Ar, and based on the VHS molecular model with parametersx ¼ 0:734; d ¼ 3:595 � 10�10 m at T ¼ 1000 K that provides the best fit in the temperature range from 1000 K to 10000 K for Ar viscosity found in Ref. [29] based on the IApotential. q1 ¼ ð1=2Þmn1U3

1 . Values in parenthesis give the difference with respect to corresponding values from the previous column.

Ref. [18], VHS x ¼ 0:734;d ¼ 3:595 A This work, VHS, x ¼ 0:734; d ¼ 3:595 A This work, AI Ar This work, VHS x ¼ 0:686; d ¼ 3:381 A

CD 1.495 1.504 (0.6%) 1.506 (0.1%) 1.508 (0.1%)CQ - 3.368 3.394 (0.7%) 3.410 (0.5%)q0=q1 237 249 (5.1%) 252 (1.2 %) 254 (0.8%)


the stagnation point is worse, presumably because the cell size inour simulations is larger that the local mean free path in the vicin-ity of the stagnation point.

Systematic comparison of the VHS molecular model and the AIpotential method is out of the scope of the present paper. We per-formed, however, a comparison of results obtained based on theVHS model and AI potential method under conditions consideredin Ref. [18] at Kn1ðHSÞ ¼ 0:05 (Table 3). This comparison showedthat values of CD and CQ found in simulations with identicalnumerical parameters based on the VHS model and AI potentialare different in less that 1%. The difference between simulationsbased on the VHS model and AI potential can be further reducedby using parameters of the VHS model that provides better fit tothe AI viscosity at elevated temperatures. In particular, by theleast-square fitting of Ar viscosity data in Ref. [29] in the temper-ature range from 1000 K to 10000 K we found the viscosity index0:686 and reference diameter 3:381 � 10�10 m at temperature of1000 K. The simulations performed with these parameters of theVHS model produce slightly smaller error compared to the AIpotential than the parameterization adopted in Ref. [18] (last col-umn in Table 3).

Fig. 4. Distributions of normal, pnw=p1 , and tangential, psw=p1 , stresses (a) and heat flux dred solid curves), AI Ar (green dashed curves), and IA He (blue dash-dotted curves) atMa1T1 ¼ 300 K. Curves for different gases visually coincide with each other in panel (a). p1figure legend, the reader is referred to the web version of this article.)

6. DSMC simulations of subsonic flows at Ma‘ ¼ 0:5

The comparison of simulations performed for the AI He, AI Ar,and HS gas in subsonic flow at Ma1 ¼ 0:5; Tref ¼ T1 ¼ 300 K, whend1ðHSÞ ¼ d1, showed that under these conditions the density, veloc-ity, and pressure field obtained for all three gases are nearly iden-tical with the local differences smaller than the accuracy achievedin our numerical simulations. For example, distributions of normaland tangential stresses at the cylinder surface obtained for differ-ent gases visually coincide with each other in Fig. 4(a). The onlyquantity that exhibits a measurable variation is the heat fluxshown in Fig. 4(b), which demonstrates a maximum difference inthe order of a few percents in the vicinity of the stagnation point,but again is nearly identical for all gases on the rest of the cylindersurface. All curves in Fig. 4 are obtained without smoothing, so thatthe noise in curves in panel (b) represents the statistical fluctua-tions of the flow parameters, which is difficult to eliminate com-pletely in DSMC simulations of subsonic flows. The difference inthe drag coefficients between the AI He, AI Ar, and HS gas is below1%, i.e. smaller than the error level adopted in the present work(Table 4). The maximum error in the heat flux coefficient calcu-

ensity, qw=q1 , (b) at the cylinder surface obtained for the HS gas (Tref ¼ T1 ¼ 300 K,¼ 0:5; d1 ¼ 1, and Tw=T1 ¼ 1. Simulations for the AI He and AI Ar are performed at

¼ n1kBT1; q1 ¼ ð1=2Þmn1U31 . (For interpretation of the references to colour in this

Table 4Drag, CD , and heat flux, CQ , coefficients obtained for the HS gas (Tref ¼ T1), AI He, and AI Ar at Ma1 ¼ 0:5; Tw=T1 ¼ 1, and T1 ¼ 300 K versus d1 . DD;Gas and DQ ;Gas are errors of theHS model given by Eqs. (27). The case of d1 ¼ 0 corresponds to free molecular flow, Eqs. (23)–(25).

d1 CD CQ

HS AI He DD;He;% AI Ar DD;Ar;% HS AI He DQ ;He;% AI Ar DQ ;Ar;%

0 9.348 9.348 0 9.348 0 0.0812 0.0812- 0 0.0812 00.1 8.26 8.253 �0.08 8.259 �0.01 0.0695 0.0699 0.58 0.0703 1.150.3 6.995 6.985 �0.14 6.979 �0.23 0.0533 0.0541 1.50 0.0545 2.251 4.882 4.884 0.04 4.834 �0.98 0.03075 0.03123 1.56 0.0318 3.413 3.197 3.201 0.13 3.203 0.19 0.0176 0.0176 0.00 0.0178 1.1410 2.059 2.061 0.10 2.063 0.19 0.00989 0.00986 �0.30 0.00992 0.30

Table 5Drag, CD , and heat flux, CQ , coefficients obtained for the HS gas (Tref ¼ T1), AI He, and AI Ar at Ma1 ¼ 2; Tw=T1 ¼ 1, and T1 ¼ 300 versus d1 . DD;Gas and DQ ;Gas are errors of the HSmodel given by Eqs. (27).

d1 CD CQ


0 3.194 3.194 0 3.194 0 0.5159 0.5159 0 0.5159 00.1 2.904 2.928 0.83 2.947 1.48 0.448 0.454 1.34 0.458 2.230.3 2.564 2.607 1.68 2.628 2.50 0.363 0.377 3.86 0.382 5.231 2.115 2.153 1.80 2.176 2.88 0.247 0.258 4.45 0.263 6.483 1.829 1.849 1.09 1.860 1.69 0.158 0.164 3.80 0.1667 5.5110 1.641 1.651 0.61 1.658 1.04 0.0938 0.097 3.41 0.0984 4.9030 1.509 1.514 0.33 1.522 0.86 0.0570 0.0585 2.63 0.0596 4.56

Fig. 5. Drag CD;Ar (squares) and heat flux CQ ;Ar (triangles) coefficients versus freestream temperature T1 calculated for the AI Ar at Ma1 ¼ 2; d1 ¼ 10, andTw=T1 ¼ 1.


lated for the HS gas is 1.6% with respect to the AI He and 3.4% withrespect to the AI Ar. For both Ar and He, the maximum error in CQ

is realized at d1 ¼ 1, i.e. in the transitional flow regime. Values oferrors of the HS model in Table 4 are calculated as follows

DD;Gas ¼ CD;Gas � CD;HS

CD;HS� 100; ð27aÞ

DQ ;Gas ¼ CQ ;Gas � CQ ;HS

CQ ;HS� 100; ð27bÞ

where Gas ¼ Ar or He.The presented results show that the HS model, where the HS

diameter is given by Eq. (19) at Tref ¼ T1, is capable of accuratepredicting the drag and heat flux coefficients of a circular cylinderin subsonic cross-flow. The choice of Tref ¼ T1 ensures calculationsof the drag force with errors below 1%, and calculations of the heatflux with errors of a few percents depending on the gas species.Thus, subsonic flows over bodies are only marginally affected bythe peculiarities of the monatomic gas species and the dependen-cies of the drag force and heat flux on flow parameters can be rep-resented in the traditional for gas dynamics form given by Eqs. (11)and (17).

7. DSMC simulations of supersonic flows at Ma‘ ¼ 2

In the supersonic flow at Ma1 ¼ 2 and Tw ¼ T1 ¼ 300 K, thedrag and heat flux coefficients obtained for the AI Ar and AI Heat the same d1 are different in less 1% and 2%, correspondingly(Table 5). Calculated values of CD and CQ given by Eqs. (16b) arefound to be weak functions of T1 (Fig. 5), so that for practical pur-poses one can assume that both CD and CQ obtained with the AIpotential method are defined by only three criteria of similarityincluding Ma1; d1, and Tw=T1. Under the conditions consideredin Fig. 5, variation of T1 at Tw=T1 ¼ 1 results in the variation ofCD and CQ within 1%.

At Ma1 ¼ 2, the HS model with the simple choice of Tref ¼ T1provides worse agreement with the results obtained for the AI Arand AI He than in the case of subsonic flow, however, the errorsin CD and CQ remain smaller than 3% and 6.5%, correspondingly,in the whole considered range of d1. The difference between the

AI potential method and HS model for Ar is usually larger than thatfor He. This conclusion also holds for subsonic flows considered inSection 6 and hypersonic flows considered below in Section 8. Sim-ilarly to subsonic flows, the maximum difference between resultsobtained with the AI potential method and HS model is observedin the transitional flow regime at d1 � 1.

The results of simulations also reveal a good agreementbetween the AI potential method and HS model at Tref ¼ T1 interms of local flow parameters. This agrement is illustrated by dis-tributions of the gas number density, velocity, and temperaturealong the stagnation stream line shown in Fig. 6 and distributionsof stresses and heat flux density along the cylinder surface shownin Fig. 7. Temperature and velocity predicted by the HS model exhi-bit maximum deviation from the distributions obtained with the AIpotential method inside the bow shock and boundary layer (Fig. 6),i.e. in the regions, where the temperature variation is steep. The


maximum difference between the stresses and heat flux densitiespredicted by the AI potential method and HS model at the cylindersurface is observed in the vicinity of points, where stress or heatflux density is maximum, i.e. at the stagnation point for the normalstress and heat flux density and at h � 45� for the tangential stress.Under conditions Tw=T1 ¼ 1 and Tref ¼ T1, the HS model tends tounderestimate the maximum values of stresses and heat flux at allconsidered d1. Correspondingly, the HS model also underestimatesthe drag and heat flux coefficients.

The presented results show that for applications, where theacceptable tolerance in determination of CD and CQ is within afew percents, calculations of drag and heat transfer of blunt bodiesin supersonic flows of a rarefied gas at Ma1 K2 can be performedwith the HS model, if the HS diameter dHS is given by Eq. (19) atTref ¼ T1. Although such accuracy can be sufficient for themajority of applications (since the further increase in the accuracyof modelling of real flows requires refining of the gas-surfaceinteraction model), it is reasonable to pose a question about anoptimum choice of Tref , which provides the minimum differencebetween the AI potential method and HS model. In order to answerthis question, we performed a series of DSMC simulationsbased on the HS model for Ar at Ma1 ¼ 2; d1 ¼ 10, and

Fig. 6. Distributions of gas number density n=n1 (a), velocity vx=U1 (b), and temperaturecurves), and IA He (blue dash-dotted curves) at Ma1 ¼ 2; d1 ¼ 10, and Tw=T1 ¼ 1. Sinterpretation of the references to colour in this figure legend, the reader is referred to

Tw=T1 ¼ 1, where Tref varies around T1. The calculated values ofCD and CQ were compared with predictions obtained for the AIAr in identical conditions, i.e. identical values of n1;U1; T1,and Tw. The results of this comparison in terms of relativeerrors of the HS model, DD ¼ ðCD;HS � CD;ArÞ=CD;Ar � 100% andDQ ¼ ðCQ ;HS � CQ ;ArÞ=CQ ;Ar � 100%, are shown in Fig. 8. For both thedrag and heat flux coefficients, the zero difference between theAI potential method and HS model is observed at approximatelythe same reference temperature Tref � 430 K. The value of Tref

strongly affects the prediction of CQ with the HS model, while CD

exhibits a relatively weak dependence on Tref : The error in CD

remains within 1%, when Tref varies in the range from 300 K to700 K.

For the better agreement with results obtained with the AIpotential method, the HS diameter in supersonic flows should cor-respond to the reference temperature that is higher than T1. Itoccurs because the distributions of stresses and heat flux densityat the cylinder surface are predominantly defined by the distribu-tions of flow parameters within the shock layer and in the bound-ary layer, where gas temperature is higher than T1. In order toaccount for this fact one can define an ‘‘optimum” Tref for the HSmodel as a characteristic temperature in the flow behind the

T=T1 (c) obtained for the HS gas (Tref ¼ 300 K, red solid curves), AI Ar (green dashedimulations based on the AI potential method are performed at T1 ¼ 300 K. (Forthe web version of this article.)

Fig. 7. Distributions of normal, pnw=p1 , and tangential, psw=p1 , stresses (a,c) and heat flux density, qw=q1 , (b,d) at the cylinder surface obtained for the HS gas(Tref ¼ T1 ¼ 300 K, red solid curves), AI Ar (green dashed curves), and IA He (blue dash-dotted curves) at Ma1 ¼ 2; Tw=T1 ¼ 1; d1 ¼ 1 (a,b), and d1 ¼ 10 (c,d). Simulationsbased on the AI potential method are performed at T1 ¼ 300 K. p1 ¼ n1kBT1; q1 ¼ ð1=2Þmn1U3

1 . (For interpretation of the references to colour in this figure legend, thereader is referred to the web version of this article.)

Fig. 8. Errors DD ¼ ðCD;HS � CD;ArÞ=CD;Ar � 100% and DQ ¼ ðCQ ;HS � CQ ;ArÞ=CQ ;Ar � 100%in the drag (squares) and heat flux (triangles) coefficients obtained with the HSmodel compared to the AI Ar versus the reference temperature Tref .Ma1 ¼ 2; d1 ¼ 10; Tw=T1 ¼ 1, and T1 ¼ 300 K. Horizontal line marks the zero levelof error; gray rectangle marks the range of optimum Tref between points whereDQ ¼ 0 and DD ¼ 0; vertical lines mark temperatures T� and T�� calculated with Eqs.(30) and (31).


bow shock. Since the accurate evaluation of the heat flux with theHS model is more difficult than calculation of the drag force andsince the total heat flux on the body surface is dominated by thelocal heat flux in the vicinity of the stagnation point, it is reason-able to consider two characteristic temperatures realized at thestagnation stream line: Temperature behind the bow shock T2

T2 ¼ T12c

cþ 1Ma21 � c� 1

cþ 1

� �2

cþ 11

Ma21þ c� 1cþ 1

!; ð28Þ

and the adiabatic stagnation temperature T0, which is realized incontinuum flows at the top of the boundary layer

T0 ¼ T1 1þ c� 12

Ma21

� �: ð29Þ

Assuming that stresses and heat flux density at the cylinder sur-face are determined by the structure of the whole shock layer, theoptimum value T� of Tref can be estimated as an algebraic mean ofTw and T2:

T� ¼ Tw þ T2

2

¼ T12

Tw

T1þ 2c

cþ 1Ma2

1 � c� 1cþ 1

� �2

cþ 11

Ma21þ c� 1cþ 1

!" #:

ð30Þ


If one assumes that the characteristic temperature of shocklayer is less important, and stresses and heat flux density at thecylinder surface are dominated by the boundary layer only, thenthe optimum value T�� of Tref can be estimated as an algebraicmean of Tw and T0:

T�� ¼ Tw þ T0

2¼ T1

2Tw

T1þ 1þ c� 1

2Ma21

� �: ð31Þ

For the flow conditions illustrated with Fig. 8, calculations withEqs. (28)–(31) result in T2 ¼ 623:4 K, T0 ¼ 700 K, T� ¼ 461:7 K, andT�� ¼ 500 K. These values of T2 and T0 are in a good agreement withthe temperature behind the shock and maximum temperature inthe shock layer observed in the DSMC simulations at sufficientlylarge d1, e.g., d1 ¼ 10 (Fig. 6(c)). The comparison of calculated val-ues of T� and T�� with the optimum Tref ¼ 340 K determined basedon results in Fig. 8 shows that Eq. (30) provides better approxima-tion for the optimum Tref than Eq. (31). Thus, in order to obtainaccurate approximations of the drag force and heat flux in the

Fig. 9. Distributions of gas temperature T=T1 obtained for the HS gas (soMa1 ¼ 10; Tw=T1 ¼ 2:5; T1 ¼ 200 K, and d1 ¼ 10 (a) and d1 ¼ 1 (b). Two solid curveonline) and Tref ¼ 3000 K (black online). (For interpretation of the references to colour i

Fig. 10. Distributions of normal, pnw=p1 , and tangential, psw=p1 , stresses (a) and heat fluxAr (dashed curves), and IA He (dash-dotted curves) at Ma1 ¼ 10; d1 ¼ 1; Tw=T1 ¼ 2:5,calculated at Tref ¼ T1 (red online) and Tref ¼ 3000 K (black online). p1 ¼ n1kBT1; q1 ¼ ðreader is referred to the web version of this article.)

DSMC simulations of supersonic flows based on the HS model,the HS diameter must be chosen based on Eq. (19) where the ref-erence temperature is defined by Eq. (30). In other words, temper-ature Tref ¼ T� must be used in order to map d1 into d1ðHSÞ inaccordance with Eq. (20) and finally predict values of CD and CQ

based on Eqs. (21). As shown in Section 8, this semi-empirical rulealso ensures satisfactory approximation of the drag force and heatflux in the DSMC simulations of hypersonic flows, where Ma1 is ashigh as 10. It is worth noting, however, that in the simulationsbased on the HS model with Tref – T1 d1ðHSÞ is different from d1.The relationship between d1ðHSÞ and d1 is given by Eq. (20) andfor argon it is presented in Fig. 2.

8. DSMC simulations of hypersonic flows at Ma‘ ¼ 10

In the case of hypersonic flows, the difference in local distribu-tions of gas parameters in reduced units (Figs. 9 and 10) as well asin the drag and heat flux coefficients (Table 6) between the AI He

lid curves), AI Ar (dashed curves), and IA He (dash-dotted curves) ats in every panel correspond to the HS gas flows calculated at Tref ¼ 200 K (redn this figure legend, the reader is referred to the web version of this article.)

density, qw=q1 , (b) at the cylinder surface obtained for the HS gas (solid curves), AIand T1 ¼ 200 K. Two solid curves in every panel correspond to the HS gas flows1=2Þmn1U3

1 . (For interpretation of the references to colour in this figure legend, the

Table 6Drag, CD , and heat flux, CQ , coefficients obtained for the HS gas (Tref ¼ T1), AI He, and AI Ar at Ma1 ¼ 10; Tw=T1 ¼ 2:5, and T1 ¼ 200 K versus d1 . DD;Gas and DQ ;Gas are errors of theHS model given by Eqs. (27).

d1 CD CQ


0 2.259 2.259 0 2.259 0 10.323 10.323 0 10.323 00.1 2.034 2.114 3.93 2.116 4.03 8.855 9.385 5.99 9.413 6.300.3 1.889 1.985 5.08 1.988 5.24 7.566 8.338 10.20 8.393 10.931 1.707 1.818 6.50 1.822 6.74 5.601 6.622 18.23 6.715 19.893 1.535 1.630 6.19 1.638 6.71 3.749 4.667 24.49 4.768 27.1810 1.414 1.461 3.32 1.468 3.82 2.347 2.895 23.35 2.980 26.9730 1.335 1.360 1.87 1.365 2.25 1.449 1.778 22.71 1.839 26.92

Fig. 11. Relative differences DD;Gas ¼ ðCD;Gas � CD;HSÞ=CD;HS � 100% and DQ ;Gas ¼ ðCQ ;Gas � CQ ;HSÞ=CQ ;HS � 100 % between the drag (a) and heat flux (b) coefficients obtained for theAI He (squares) or AI Ar (triangles) and HS gas at Ma1 ¼ 10; d1 ¼ 10, and Tw=T1 ¼ 2:5. The HS diameter is determined at Tref ¼ 200 K (bold symbols) or Tref ¼ 3000 K (opensymbols).

Fig. 12. Errors ðCD;HS � CD;ArÞ=CD;Ar � 100% and ðCQ ;HS � CQ ;ArÞ=CQ ;Ar � 100% in thedrag (squares) and heat flux (triangles) coefficients obtained with the HS modelcompared to the AI Ar versus reference temperature Tref . Ma1 ¼ 10; d1 ¼ 10, andTw=T1 ¼ 2:5 and T1 ¼ 200 K.


and AI Ar is found to be small for all considered cases. For instance,the difference in the reduced heat flux q=q1 at the stagnationpoint is 1:5%. Thus, even in hypersonic flows, the actual depen-dence of simulation results on the monatomic gas species ismarginal.

At the same time, the use of the HS model with Tref ¼ T1 resultsin the pronounced errors in distributions of local gas parameters,(red solid curves in Figs. 9 and 10). In particular, the HS modelstrongly overestimates the maximum gas temperature at d1 ¼ 1(Fig. 9(b)). The difference in the reduced heat flux density q=q1at the stagnation point between the AI Ar and HS gas is 17:5%.The difference in the drag and heat flux coefficients rises up to7% and 27%, correspondingly. The maximum difference in CD andCQ between the AI Ar and HS gas occurs in the transitional flowregime at 1 6 d1 6 3 (Fig. 11). With further increase in d1, the dif-ference in CD tends to decrease relatively fast, while error in CQ

remains approximately constant when d1 varies from 3 to 30.The HS model poorly approximates the results obtained with

the AI potential method at Ma1 ¼ 10 and Tref ¼ T1, because gastemperature in the shock layer in this case is in order of magnitudehigher than T1, so that T1 cannot be considered as a characteristicflow temperature. The choice of Tref ¼ T1 results in the overesti-mation of the HS diameter and ‘‘exaggeration” of all effects inducedby the interatomic collisions, so that distributions of gas parame-ters obtained with the HS model at Tref ¼ T1 become ‘‘closer” tothe limit of continuum flows than their counterparts obtained withthe AI potential method. This fact is clearly seen in Fig. 9(a). Thedistribution of temperature obtained with the HS model is muchsteeper at the bow shock and contains a flat ‘‘plateau” correspond-ing to the part of the shock layer between the bow shock andboundary layer. In the flows of the AI He and AI Ar, however, thebow shock and boundary layer merge into a single flow structuretypical for the transitional flow regime.

In order to find the optimum value of Tref we performed DSMCsimulations based on the HS model for Ar with varying Tref . Theobtained relative differences in the drag and heat flux coefficientscompared to results found with the AI potential method are shownin Fig. 12. The errors in CD and CQ drop to zero at approximately thesame Tref ¼ 2500 K. The temperatures given by Eqs. (30) and (31)at Ma1 ¼ 10 and Tw=T1 ¼ 2:5 are equal to T� ¼ 3462 K andT�� ¼ 3683 K and, thus, both equations somewhat overestimate

Fig. 13. Fields of temperature T=T1 obtained at Ma1 ¼ 10; d1 ¼ 10, andTw=T1 ¼ 2:5: (a), AI Ar; (b), HS gas at Tref ¼ 200 K; and (c), HS gas at Tref ¼ 3000 K.


the optimum Tref . At the same time this overestimation is notimportant from the practical point of view, because the choice ofTref ¼ T� results in only 0:3% error in CD and 1:7% in the heat flux.Moreover, the optimum value of Tref is not a well-defined quantity,since it also depends on d1. Thus, the rule Tref ¼ T� can be recom-mended for DSMC simulations of super- and hypersonic flows of ararefied gas based on the HS model.

In order to illustrate the effect of an increase in Tref on the localflow parameters as well as on CD and CQ a series of simulations was

Table 7Drag, CD , and heat flux, CQ , coefficients obtained for the HS gas (Tref ¼ 3000 K), AI He, and AIthe HS model given by Eqs. (27).

d1 d1ðHSÞ CD

HS AI Gas

Argon0.1 0.0527 2.109 2.1160.3 0.158 1.973 1.9881 0.527 1.809 1.8223 1.58 1.630 1.63810 5.27 1.470 1.46830 15.8 1.369 1.365

Helium0.1 0.0557 2.104 2.1140.3 0.167 1.967 1.9851 0.557 1.800 1.8183 1.67 1.622 1.63010 5.57 1.464 1.46130 16.7 1.365 1.360

performed at Tref ¼ 3000 K. The results of these simulations ford1 ¼ 1 and 10 are shown in Figs. 9 and 10 by black solid curves.At this Tref , distributions of temperature along the stagnationstreamline and heat flux density along the cylinder surfaceobtained with the HS model closely follow the distributionsobtained for the AI Ar, and the distributions of stresses are practi-cally identical. This choice of Tref also ensures accurate predictionof the global temperature field, including the low-temperaturewake zone (Fig. 13). The error in CD at this Tref is within 1% for bothAr and He, while the error in CQ is limited by 3% for all cases con-sidered (Fig. 11 and Table 7). The simulations with the HS model atTref ¼ 3000 K were performed separately for Ar and He, since dHS

depends on lAIðTref Þ and, thus, d1ðHSÞ=d1 for the HS model approx-imating the AI Ar and AI He are different at the same Tref . The dif-ference in d1ðHSÞ=d1 between Ar and He, however, is relativelysmall, and the errors in CD and CQ obtained with the HS modelfor Ar and He at fixed d1 vary within 1%.

9. Discussion and conclusions

The ab initio (AI) potential method, based on quantum–mechan-ical calculations of pairwise interatomic potentials, is used for sam-pling of inter-particle collisions in DSMC simulations of Ar and Heflows past a circular cylinder in the ranges of the free stream Machnumber Ma1 from 0.5 to 10, and rarefaction parameters d1 from0.1 to 30, which span the range of conditions from nearlyfree-molecular to nearly continuum flows. It is found that the AIpotential method can be efficiently used for kinetic simulationsof two-dimensional rarefied monatomic gas flows over bodies. Thismethod does not require any fitting parameter in contrast to thepopular HS and VHS molecular models and can completely replacethe latter if the potentials are known for atomic pairs under consid-eration. The results based on the AI potential method can be usedas benchmark data to test new numerical methods and models forrarefied gas flows.

The results obtained based on the AI potential method and rep-resented in reduced units, including the drag and heat flux coeffi-cients, are not fully scaled with the Mach number, rarefactionparameter, and free-stream-to-surface temperature ratio, but alsoare functions of the adopted interatomic interaction potentialand, thus, depend on the gas species under consideration anddimensional temperature scale, e.g., free stream temperature. Thecomparison of the drag and heat flux coefficients obtained basedon the AI potential method for Ar and He revealed, however, only

Ar at Ma1 ¼ 10; Tw=T1 ¼ 2:5, and T1 ¼ 200 K versus d1 . DD;Gas and DQ ;Gas are errors of

CQ

DD;Gas;% HS AI Gas DQ ;Gas ;%

0.33 9.397 9.413 0.170.76 8.365 8.393 0.330.72 6.713 6.715 0.030.49 4.783 4.768 �0.31�0.13 3.011 2.980 �1.03�0.29 1.871 1.839 �1.71

0.48 9.360 9.385 0.270.92 8.308 8.338 0.361.00 6.624 6.622 �0.030.49 4.691 4.667 �0.51�0.20 2.943 2.895 �1.63�0.37 1.825 1.778 �2.58

Table 8Ratio of CPU times sAI=sHS required for collision sampling based on the AI potentialmethod and HS model at fixed values of K that defines the majorant ½rcr max (seedetails in the text).

Flow conditions sAI=sHS at fixed K

K ¼ 6 K ¼ 60 K ¼ 300

Ma1 ¼ 2; d1 ¼ 10 1.35 1.58 1.15Ma1 ¼ 10; d1 ¼ 10 1.26 1.56 1.04


marginal dependence of these coefficients on the gas species andfree stream temperature.

The results of the DSMC simulations obtained with the AIpotential method were systematically compared with the resultsobtained under identical conditions based on the HS model. Thediameter of the HS molecules was chosen based on the conditionof the same viscosity of the AI and HS gases at some reference tem-perature. In sub- and supersonic flows at moderate free streamMach number, Ma1 6 2, the simple choice of the reference tem-perature equal to the free stream temperature ensures calculationsof the drag and heat flux coefficients based on the HS model witherrors of 3%. In hypersonic flows at Ma1 ¼ 10 such a choice of thereference temperature results in larger errors in local flow param-eters as well as drag and heat flux coefficients. This is explained bythe fact that in hypersonic flows the free stream temperature is nolonger the characteristic gas temperature in the shock layer. Anunderestimation of the reference temperature results in the strongoverestimation of the collision effects in simulations with the HSmodel in comparison to simulations based on the AI potentials.

With the choice of the reference temperature in the form of anaverage between the surface temperature and temperature behindthe bow shock given by Eq. (30) one can obtain a satisfactoryagreement between the AI potential method and HS model interms of local flow parameters. For all flow conditions consideredin the present paper, this choice of the reference temperaturereduces the error in the drag and heat flux coefficients calculatedbased on the HS model to 1% and 3%, respectively. This rule allowsone to map, in accordance with Eq. (20), the density parameter forthe real gas flow d1 into the density parameter d1ðHSÞ that must beused for simulations of corresponding flow with the HS model.Thus, with the proper choice of the molecular diameter, the simpleand robust HS molecular model is capable of predicting the rar-efied gas flows past blunt bodies with practically acceptableaccuracy.

Acknowledgments

Authors thank M. Yu. Plotnikov for providing data for compar-ison of the simulation results. A.V. acknowledges the support ofthe present work by the NSF CAREER award CMMI-1554589. F.S.thanks CNPq, Brazil for the support of his research (grant303697/2014-8). Computational support is provided by the Ala-bama Supercomputer Center.

Appendix A. Comparison of the CPU time required for collisionsampling based on the AI potential method and HS model

The CPU time required for collision sampling in the DSMCmethod based on the AI potential method, sAI , is longer than theCPU time needed for collision sampling based on the HS model,sHS. The value of sAI=sHS is a subject of large variability and dependson both flow conditions and peculiarities of the numericalapproach used to sample values of the deflection angle v. In orderto analyze sAI=sHS, the calculations were performed as follows.First, the parallel computational code was supplemented with cal-culations of the CPU time spent by every parallel process for execu-tion of the collision sampling routine. Second, the calculations ofthe CPU time were performed for both the HS model and AI poten-tial method staring from the same initial distributions of particlesin the computational domain, which were previously obtained andanalysed in Sections 7 and 8 for AI He at d1 ¼ 10 and Ma1 ¼ 2 orMa1 ¼ 10. At every run of the code, sAI or sHS was defined for cer-tain parallel process as an average value for 50 sequential timesteps. For every set of parameters, final values of sAI or sHS weredetermined as minimum values among five independent runs.

There are two major reasons that make sAI larger than sHS. Thefirst reason is the difference in calculations on the post-collisionvelocities according to Eq. (3). In the HS model, all directions ofvector d0 are equally probable, while in the AI potential methodone needs to select first a random value of b, then find correspond-ing vðEr ; bÞ by interpolating the tabulated values of the deflectionangle, and finally find d0 using Eq. (4). In the NTCmethod of the col-lision sampling, it is necessary to introduce a majorant ½rcrmax thatmust be larger than rðcrÞcr for any colliding molecular pair. Inevery our simulation, the value of the majorant was chosen to beconstant and defined as ½rcr max ¼ KrC1, where r is equal to eitherrHS or rAI and K is the adjustable constant that was varied from 6 to300. The values of sAI=sHS in Table 8 are obtained at the same K forboth the HS and AI gases and characterize the extra CPU timerequired for sampling of post-collision velocities with the AI poten-tial method. This extra time does not exceed 60% of sHS, but typi-cally is about 25% of sHS. A comparison of CPU time for the HSgas and AI potentials was also performed in Ref. [4]. It was shownthat calculations based on the AI potential method take about 40%more CPU time than those for the HS gas. This is consistent withdata in Table 8.

The second reason is related to the fact that accurate calcula-tions with the AI potential method require bmax dHS, becausethe effect of low-energy collisions with large b may not be small.It results, in general, in larger value of ½rcr max and sampling of lar-ger number of collisions compared to simulations based on the HSmodel. It was found that simulations with the HS model can beperformed with K ¼ 6 at Ma1 ¼ 2 and K ¼ 60 at Ma1 ¼ 10, whilesimulations based on the AI potential method are accurate atK ¼ 60 and K ¼ 300, correspondingly. We estimated sAI and sHSin simulations with minimum available K for every approach andfound then that sAI=sHS ¼ 6:44 at Ma1 ¼ 2 and sAI=sHS ¼ 3:46 atMa1 ¼ 10. In our parallel simulations, however, the CPU time forcollisions sampling was not larger than 50% of the total wall time,so the overall wall time in simulations based on the AI potentialmethod was longer than the wall time in corresponding HS simu-lations in less than 3 times.

In addition, we compared the total wall time in simulationsbased on the VHS model and AI potential method under conditionsconsidered in Ref. [18]. These simulations were discussed in theend of Section 5 and in Table 3. It was found that the wall timein simulations based on the AI potential method is in 14% longercompared to simulations based on the VHS model.

It is worth noting that we used the algorithmically simplest forcollision sampling in the AI potential method. For example, one canuse an uneven table of vðEr; bÞ where bmax is not a constant, butdecreases with increasing Er . With such a table, the required valueof the majorant ½rcr max and the CPU time for collision sampling canbe substantially smaller. The use of individual values of ½rcr max inevery cell of the computational mesh can further decrease sAI=sHS.

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